LMFDB - Embedded newform 6825.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6825,2,Mod(1,6825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6825.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6825.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$54.4978993795$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1365)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	6825.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -4.80423 q^{11} +0.618034 q^{12} +1.00000 q^{13} -1.61803 q^{14} -4.85410 q^{16} +3.24669 q^{17} -1.61803 q^{18} -0.891491 q^{19} +1.00000 q^{21} +7.77340 q^{22} +1.41164 q^{23} +2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} +0.618034 q^{28} +0.459650 q^{29} +0.340519 q^{31} +3.38197 q^{32} -4.80423 q^{33} -5.25325 q^{34} +0.618034 q^{36} -1.71592 q^{37} +1.44246 q^{38} +1.00000 q^{39} -3.07768 q^{41} -1.61803 q^{42} -11.8885 q^{43} -2.96917 q^{44} -2.28408 q^{46} -4.50953 q^{47} -4.85410 q^{48} +1.00000 q^{49} +3.24669 q^{51} +0.618034 q^{52} -0.405074 q^{53} -1.61803 q^{54} +2.23607 q^{56} -0.891491 q^{57} -0.743729 q^{58} +8.94241 q^{59} -10.3138 q^{61} -0.550972 q^{62} +1.00000 q^{63} +4.23607 q^{64} +7.77340 q^{66} +8.24367 q^{67} +2.00656 q^{68} +1.41164 q^{69} +7.57763 q^{71} +2.23607 q^{72} +2.48276 q^{73} +2.77642 q^{74} -0.550972 q^{76} -4.80423 q^{77} -1.61803 q^{78} -9.07112 q^{79} +1.00000 q^{81} +4.97980 q^{82} -3.83035 q^{83} +0.618034 q^{84} +19.2360 q^{86} +0.459650 q^{87} -10.7426 q^{88} +15.2935 q^{89} +1.00000 q^{91} +0.872441 q^{92} +0.340519 q^{93} +7.29657 q^{94} +3.38197 q^{96} -10.4893 q^{97} -1.61803 q^{98} -4.80423 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 6 q^{16} - 2 q^{17} - 2 q^{18} - 2 q^{19} + 4 q^{21} + 2 q^{22} - 8 q^{23} - 2 q^{26} + 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 6 * q^16 - 2 * q^17 - 2 * q^18 - 2 * q^19 + 4 * q^21 + 2 * q^22 - 8 * q^23 - 2 * q^26 + 4 * q^27 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 18 * q^32 - 4 * q^33 - 4 * q^34 - 2 * q^36 - 10 * q^37 + 6 * q^38 + 4 * q^39 - 2 * q^42 - 18 * q^43 + 2 * q^44 - 6 * q^46 - 12 * q^47 - 6 * q^48 + 4 * q^49 - 2 * q^51 - 2 * q^52 + 10 * q^53 - 2 * q^54 - 2 * q^57 + 8 * q^58 + 4 * q^59 - 20 * q^61 - 4 * q^62 + 4 * q^63 + 8 * q^64 + 2 * q^66 - 10 * q^67 + 6 * q^68 - 8 * q^69 - 14 * q^71 - 14 * q^73 + 10 * q^74 - 4 * q^76 - 4 * q^77 - 2 * q^78 - 26 * q^79 + 4 * q^81 - 12 * q^83 - 2 * q^84 + 4 * q^86 - 6 * q^87 + 20 * q^89 + 4 * q^91 + 14 * q^92 - 2 * q^93 + 16 * q^94 + 18 * q^96 - 16 * q^97 - 2 * q^98 - 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.80423	−1.44853	−0.724264	−	0.689522i	$-0.757820\pi$
$11$	−4.80423	−1.44853	−0.724264	+	0.689522i	$0.757820\pi$
$12$	0.618034	0.178411
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	3.24669	0.787438	0.393719	−	0.919231i	$-0.371188\pi$
$17$	3.24669	0.787438	0.393719	+	0.919231i	$0.371188\pi$
$18$	−1.61803	−0.381374
$19$	−0.891491	−0.204522	−0.102261	−	0.994758i	$-0.532608\pi$
$19$	−0.891491	−0.204522	−0.102261	+	0.994758i	$0.532608\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	7.77340	1.65729
$23$	1.41164	0.294347	0.147173	−	0.989111i	$-0.452982\pi$
$23$	1.41164	0.294347	0.147173	+	0.989111i	$0.452982\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−1.61803	−0.317323
$27$	1.00000	0.192450
$28$	0.618034	0.116797
$29$	0.459650	0.0853548	0.0426774	−	0.999089i	$-0.486411\pi$
$29$	0.459650	0.0853548	0.0426774	+	0.999089i	$0.486411\pi$
$30$	0	0
$31$	0.340519	0.0611591	0.0305795	−	0.999532i	$-0.490265\pi$
$31$	0.340519	0.0611591	0.0305795	+	0.999532i	$0.490265\pi$
$32$	3.38197	0.597853
$33$	−4.80423	−0.836308
$34$	−5.25325	−0.900926
$35$	0	0
$36$	0.618034	0.103006
$37$	−1.71592	−0.282096	−0.141048	−	0.990003i	$-0.545047\pi$
$37$	−1.71592	−0.282096	−0.141048	+	0.990003i	$0.545047\pi$
$38$	1.44246	0.233998
$39$	1.00000	0.160128
$40$	0	0
$41$	−3.07768	−0.480653	−0.240327	−	0.970692i	$-0.577255\pi$
$41$	−3.07768	−0.480653	−0.240327	+	0.970692i	$0.577255\pi$
$42$	−1.61803	−0.249668
$43$	−11.8885	−1.81298	−0.906488	−	0.422232i	$-0.861247\pi$
$43$	−11.8885	−1.81298	−0.906488	+	0.422232i	$0.861247\pi$
$44$	−2.96917	−0.447620
$45$	0	0
$46$	−2.28408	−0.336769
$47$	−4.50953	−0.657782	−0.328891	−	0.944368i	$-0.606675\pi$
$47$	−4.50953	−0.657782	−0.328891	+	0.944368i	$0.606675\pi$
$48$	−4.85410	−0.700629
$49$	1.00000	0.142857
$50$	0	0
$51$	3.24669	0.454627
$52$	0.618034	0.0857059
$53$	−0.405074	−0.0556412	−0.0278206	−	0.999613i	$-0.508857\pi$
$53$	−0.405074	−0.0556412	−0.0278206	+	0.999613i	$0.508857\pi$
$54$	−1.61803	−0.220187
$55$	0	0
$56$	2.23607	0.298807
$57$	−0.891491	−0.118081
$58$	−0.743729	−0.0976563
$59$	8.94241	1.16420	0.582101	−	0.813116i	$-0.302231\pi$
$59$	8.94241	1.16420	0.582101	+	0.813116i	$0.302231\pi$
$60$	0	0
$61$	−10.3138	−1.32054	−0.660270	−	0.751028i	$-0.729558\pi$
$61$	−10.3138	−1.32054	−0.660270	+	0.751028i	$0.729558\pi$
$62$	−0.550972	−0.0699735
$63$	1.00000	0.125988
$64$	4.23607	0.529508
$65$	0	0
$66$	7.77340	0.956840
$67$	8.24367	1.00712	0.503562	−	0.863959i	$-0.332022\pi$
$67$	8.24367	1.00712	0.503562	+	0.863959i	$0.332022\pi$
$68$	2.00656	0.243332
$69$	1.41164	0.169941
$70$	0	0
$71$	7.57763	0.899299	0.449649	−	0.893205i	$-0.351549\pi$
$71$	7.57763	0.899299	0.449649	+	0.893205i	$0.351549\pi$
$72$	2.23607	0.263523
$73$	2.48276	0.290585	0.145292	−	0.989389i	$-0.453588\pi$
$73$	2.48276	0.290585	0.145292	+	0.989389i	$0.453588\pi$
$74$	2.77642	0.322752
$75$	0	0
$76$	−0.550972	−0.0632008
$77$	−4.80423	−0.547492
$78$	−1.61803	−0.183206
$79$	−9.07112	−1.02058	−0.510290	−	0.860002i	$-0.670462\pi$
$79$	−9.07112	−1.02058	−0.510290	+	0.860002i	$0.670462\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	4.97980	0.549927
$83$	−3.83035	−0.420436	−0.210218	−	0.977655i	$-0.567417\pi$
$83$	−3.83035	−0.420436	−0.210218	+	0.977655i	$0.567417\pi$
$84$	0.618034	0.0674330
$85$	0	0
$86$	19.2360	2.07427
$87$	0.459650	0.0492796
$88$	−10.7426	−1.14516
$89$	15.2935	1.62111	0.810556	−	0.585661i	$-0.199165\pi$
$89$	15.2935	1.62111	0.810556	+	0.585661i	$0.199165\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0.872441	0.0909582
$93$	0.340519	0.0353102
$94$	7.29657	0.752583
$95$	0	0
$96$	3.38197	0.345170
$97$	−10.4893	−1.06503	−0.532515	−	0.846421i	$-0.678753\pi$
$97$	−10.4893	−1.06503	−0.532515	+	0.846421i	$0.678753\pi$
$98$	−1.61803	−0.163446
$99$	−4.80423	−0.482843
$100$	0	0
$101$	2.46151	0.244930	0.122465	−	0.992473i	$-0.460920\pi$
$101$	2.46151	0.244930	0.122465	+	0.992473i	$0.460920\pi$
$102$	−5.25325	−0.520150
$103$	−4.66906	−0.460056	−0.230028	−	0.973184i	$-0.573882\pi$
$103$	−4.66906	−0.460056	−0.230028	+	0.973184i	$0.573882\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	0.655423	0.0636604
$107$	−6.28949	−0.608028	−0.304014	−	0.952668i	$-0.598327\pi$
$107$	−6.28949	−0.608028	−0.304014	+	0.952668i	$0.598327\pi$
$108$	0.618034	0.0594703
$109$	−3.92045	−0.375511	−0.187756	−	0.982216i	$-0.560121\pi$
$109$	−3.92045	−0.375511	−0.187756	+	0.982216i	$0.560121\pi$
$110$	0	0
$111$	−1.71592	−0.162868
$112$	−4.85410	−0.458670
$113$	−2.43841	−0.229386	−0.114693	−	0.993401i	$-0.536588\pi$
$113$	−2.43841	−0.229386	−0.114693	+	0.993401i	$0.536588\pi$
$114$	1.44246	0.135099
$115$	0	0
$116$	0.284079	0.0263761
$117$	1.00000	0.0924500
$118$	−14.4691	−1.33199
$119$	3.24669	0.297624
$120$	0	0
$121$	12.0806	1.09824
$122$	16.6880	1.51086
$123$	−3.07768	−0.277505
$124$	0.210453	0.0188992
$125$	0	0
$126$	−1.61803	−0.144146
$127$	−13.3974	−1.18882	−0.594412	−	0.804161i	$-0.702615\pi$
$127$	−13.3974	−1.18882	−0.594412	+	0.804161i	$0.702615\pi$
$128$	−13.6180	−1.20368
$129$	−11.8885	−1.04672
$130$	0	0
$131$	−20.5593	−1.79627	−0.898137	−	0.439716i	$-0.855079\pi$
$131$	−20.5593	−1.79627	−0.898137	+	0.439716i	$0.855079\pi$
$132$	−2.96917	−0.258434
$133$	−0.891491	−0.0773021
$134$	−13.3385	−1.15227
$135$	0	0
$136$	7.25982	0.622524
$137$	−4.74258	−0.405186	−0.202593	−	0.979263i	$-0.564937\pi$
$137$	−4.74258	−0.405186	−0.202593	+	0.979263i	$0.564937\pi$
$138$	−2.28408	−0.194434
$139$	14.9069	1.26439	0.632193	−	0.774811i	$-0.282155\pi$
$139$	14.9069	1.26439	0.632193	+	0.774811i	$0.282155\pi$
$140$	0	0
$141$	−4.50953	−0.379771
$142$	−12.2609	−1.02891
$143$	−4.80423	−0.401750
$144$	−4.85410	−0.404508
$145$	0	0
$146$	−4.01719	−0.332465
$147$	1.00000	0.0824786
$148$	−1.06050	−0.0871724
$149$	−9.09372	−0.744986	−0.372493	−	0.928035i	$-0.621497\pi$
$149$	−9.09372	−0.744986	−0.372493	+	0.928035i	$0.621497\pi$
$150$	0	0
$151$	−12.9864	−1.05682	−0.528408	−	0.848991i	$-0.677211\pi$
$151$	−12.9864	−1.05682	−0.528408	+	0.848991i	$0.677211\pi$
$152$	−1.99344	−0.161689
$153$	3.24669	0.262479
$154$	7.77340	0.626399
$155$	0	0
$156$	0.618034	0.0494823
$157$	2.55703	0.204073	0.102036	−	0.994781i	$-0.467464\pi$
$157$	2.55703	0.204073	0.102036	+	0.994781i	$0.467464\pi$
$158$	14.6774	1.16767
$159$	−0.405074	−0.0321245
$160$	0	0
$161$	1.41164	0.111253
$162$	−1.61803	−0.127125
$163$	25.2136	1.97488	0.987439	−	0.157999i	$-0.0505044\pi$
$163$	25.2136	1.97488	0.987439	+	0.157999i	$0.0505044\pi$
$164$	−1.90211	−0.148530
$165$	0	0
$166$	6.19764	0.481030
$167$	−14.4068	−1.11483	−0.557414	−	0.830235i	$-0.688206\pi$
$167$	−14.4068	−1.11483	−0.557414	+	0.830235i	$0.688206\pi$
$168$	2.23607	0.172516
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.891491	−0.0681741
$172$	−7.34748	−0.560240
$173$	−10.0659	−0.765297	−0.382648	−	0.923894i	$-0.624988\pi$
$173$	−10.0659	−0.765297	−0.382648	+	0.923894i	$0.624988\pi$
$174$	−0.743729	−0.0563819
$175$	0	0
$176$	23.3202	1.75783
$177$	8.94241	0.672152
$178$	−24.7455	−1.85475
$179$	8.15910	0.609840	0.304920	−	0.952378i	$-0.401370\pi$
$179$	8.15910	0.609840	0.304920	+	0.952378i	$0.401370\pi$
$180$	0	0
$181$	−20.5509	−1.52753	−0.763767	−	0.645492i	$-0.776652\pi$
$181$	−20.5509	−1.52753	−0.763767	+	0.645492i	$0.776652\pi$
$182$	−1.61803	−0.119937
$183$	−10.3138	−0.762414
$184$	3.15652	0.232702
$185$	0	0
$186$	−0.550972	−0.0403992
$187$	−15.5978	−1.14063
$188$	−2.78704	−0.203266
$189$	1.00000	0.0727393
$190$	0	0
$191$	14.8310	1.07313	0.536567	−	0.843858i	$-0.319721\pi$
$191$	14.8310	1.07313	0.536567	+	0.843858i	$0.319721\pi$
$192$	4.23607	0.305712
$193$	−4.38447	−0.315601	−0.157801	−	0.987471i	$-0.550440\pi$
$193$	−4.38447	−0.315601	−0.157801	+	0.987471i	$0.550440\pi$
$194$	16.9721	1.21852
$195$	0	0
$196$	0.618034	0.0441453
$197$	−5.40321	−0.384963	−0.192481	−	0.981301i	$-0.561653\pi$
$197$	−5.40321	−0.384963	−0.192481	+	0.981301i	$0.561653\pi$
$198$	7.77340	0.552432
$199$	−11.5692	−0.820119	−0.410059	−	0.912059i	$-0.634492\pi$
$199$	−11.5692	−0.820119	−0.410059	+	0.912059i	$0.634492\pi$
$200$	0	0
$201$	8.24367	0.581464
$202$	−3.98281	−0.280230
$203$	0.459650	0.0322611
$204$	2.00656	0.140488
$205$	0	0
$206$	7.55470	0.526361
$207$	1.41164	0.0981157
$208$	−4.85410	−0.336571
$209$	4.28293	0.296256
$210$	0	0
$211$	−10.7653	−0.741113	−0.370556	−	0.928810i	$-0.620833\pi$
$211$	−10.7653	−0.741113	−0.370556	+	0.928810i	$0.620833\pi$
$212$	−0.250349	−0.0171941
$213$	7.57763	0.519210
$214$	10.1766	0.695659
$215$	0	0
$216$	2.23607	0.152145
$217$	0.340519	0.0231160
$218$	6.34342	0.429631
$219$	2.48276	0.167769
$220$	0	0
$221$	3.24669	0.218396
$222$	2.77642	0.186341
$223$	−9.65948	−0.646847	−0.323423	−	0.946254i	$-0.604834\pi$
$223$	−9.65948	−0.646847	−0.323423	+	0.946254i	$0.604834\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	3.94542	0.262446
$227$	27.8250	1.84681	0.923406	−	0.383825i	$-0.125393\pi$
$227$	27.8250	1.84681	0.923406	+	0.383825i	$0.125393\pi$
$228$	−0.550972	−0.0364890
$229$	8.53026	0.563695	0.281848	−	0.959459i	$-0.409053\pi$
$229$	8.53026	0.563695	0.281848	+	0.959459i	$0.409053\pi$
$230$	0	0
$231$	−4.80423	−0.316095
$232$	1.02781	0.0674789
$233$	21.1471	1.38540	0.692698	−	0.721228i	$-0.256422\pi$
$233$	21.1471	1.38540	0.692698	+	0.721228i	$0.256422\pi$
$234$	−1.61803	−0.105774
$235$	0	0
$236$	5.52671	0.359758
$237$	−9.07112	−0.589233
$238$	−5.25325	−0.340518
$239$	18.1840	1.17623	0.588113	−	0.808779i	$-0.299871\pi$
$239$	18.1840	1.17623	0.588113	+	0.808779i	$0.299871\pi$
$240$	0	0
$241$	−11.7022	−0.753803	−0.376901	−	0.926253i	$-0.623010\pi$
$241$	−11.7022	−0.753803	−0.376901	+	0.926253i	$0.623010\pi$
$242$	−19.5468	−1.25652
$243$	1.00000	0.0641500
$244$	−6.37425	−0.408069
$245$	0	0
$246$	4.97980	0.317500
$247$	−0.891491	−0.0567242
$248$	0.761425	0.0483505
$249$	−3.83035	−0.242739
$250$	0	0
$251$	6.64185	0.419230	0.209615	−	0.977784i	$-0.432779\pi$
$251$	6.64185	0.419230	0.209615	+	0.977784i	$0.432779\pi$
$252$	0.618034	0.0389325
$253$	−6.78183	−0.426370
$254$	21.6774	1.36016
$255$	0	0
$256$	13.5623	0.847644
$257$	−16.4358	−1.02524	−0.512618	−	0.858617i	$-0.671324\pi$
$257$	−16.4358	−1.02524	−0.512618	+	0.858617i	$0.671324\pi$
$258$	19.2360	1.19758
$259$	−1.71592	−0.106622
$260$	0	0
$261$	0.459650	0.0284516
$262$	33.2656	2.05516
$263$	−27.9057	−1.72074	−0.860368	−	0.509673i	$-0.829766\pi$
$263$	−27.9057	−1.72074	−0.860368	+	0.509673i	$0.829766\pi$
$264$	−10.7426	−0.661160
$265$	0	0
$266$	1.44246	0.0884431
$267$	15.2935	0.935950
$268$	5.09487	0.311219
$269$	27.3788	1.66932	0.834659	−	0.550768i	$-0.185665\pi$
$269$	27.3788	1.66932	0.834659	+	0.550768i	$0.185665\pi$
$270$	0	0
$271$	−3.10600	−0.188676	−0.0943381	−	0.995540i	$-0.530073\pi$
$271$	−3.10600	−0.188676	−0.0943381	+	0.995540i	$0.530073\pi$
$272$	−15.7598	−0.955576
$273$	1.00000	0.0605228
$274$	7.67365	0.463582
$275$	0	0
$276$	0.872441	0.0525148
$277$	30.2146	1.81542	0.907710	−	0.419599i	$-0.137829\pi$
$277$	30.2146	1.81542	0.907710	+	0.419599i	$0.137829\pi$
$278$	−24.1198	−1.44661
$279$	0.340519	0.0203864
$280$	0	0
$281$	−0.922959	−0.0550591	−0.0275296	−	0.999621i	$-0.508764\pi$
$281$	−0.922959	−0.0550591	−0.0275296	+	0.999621i	$0.508764\pi$
$282$	7.29657	0.434504
$283$	−16.7953	−0.998376	−0.499188	−	0.866494i	$-0.666368\pi$
$283$	−16.7953	−0.998376	−0.499188	+	0.866494i	$0.666368\pi$
$284$	4.68323	0.277899
$285$	0	0
$286$	7.77340	0.459651
$287$	−3.07768	−0.181670
$288$	3.38197	0.199284
$289$	−6.45901	−0.379942
$290$	0	0
$291$	−10.4893	−0.614895
$292$	1.53443	0.0897956
$293$	5.53066	0.323104	0.161552	−	0.986864i	$-0.448350\pi$
$293$	5.53066	0.323104	0.161552	+	0.986864i	$0.448350\pi$
$294$	−1.61803	−0.0943657
$295$	0	0
$296$	−3.83692	−0.223016
$297$	−4.80423	−0.278769
$298$	14.7139	0.852356
$299$	1.41164	0.0816372
$300$	0	0
$301$	−11.8885	−0.685240
$302$	21.0124	1.20913
$303$	2.46151	0.141410
$304$	4.32739	0.248193
$305$	0	0
$306$	−5.25325	−0.300309
$307$	−31.3957	−1.79185	−0.895923	−	0.444208i	$-0.853485\pi$
$307$	−31.3957	−1.79185	−0.895923	+	0.444208i	$0.853485\pi$
$308$	−2.96917	−0.169184
$309$	−4.66906	−0.265614
$310$	0	0
$311$	2.73362	0.155009	0.0775046	−	0.996992i	$-0.475305\pi$
$311$	2.73362	0.155009	0.0775046	+	0.996992i	$0.475305\pi$
$312$	2.23607	0.126592
$313$	−2.92489	−0.165325	−0.0826624	−	0.996578i	$-0.526342\pi$
$313$	−2.92489	−0.165325	−0.0826624	+	0.996578i	$0.526342\pi$
$314$	−4.13736	−0.233485
$315$	0	0
$316$	−5.60626	−0.315377
$317$	0.810148	0.0455024	0.0227512	−	0.999741i	$-0.492757\pi$
$317$	0.810148	0.0455024	0.0227512	+	0.999741i	$0.492757\pi$
$318$	0.655423	0.0367543
$319$	−2.20826	−0.123639
$320$	0	0
$321$	−6.28949	−0.351045
$322$	−2.28408	−0.127287
$323$	−2.89440	−0.161049
$324$	0.618034	0.0343352
$325$	0	0
$326$	−40.7964	−2.25950
$327$	−3.92045	−0.216801
$328$	−6.88191	−0.379990
$329$	−4.50953	−0.248618
$330$	0	0
$331$	−5.02278	−0.276077	−0.138038	−	0.990427i	$-0.544080\pi$
$331$	−5.02278	−0.276077	−0.138038	+	0.990427i	$0.544080\pi$
$332$	−2.36729	−0.129922
$333$	−1.71592	−0.0940319
$334$	23.3106	1.27550
$335$	0	0
$336$	−4.85410	−0.264813
$337$	−14.9881	−0.816455	−0.408227	−	0.912880i	$-0.633853\pi$
$337$	−14.9881	−0.816455	−0.408227	+	0.912880i	$0.633853\pi$
$338$	−1.61803	−0.0880094
$339$	−2.43841	−0.132436
$340$	0	0
$341$	−1.63593	−0.0885907
$342$	1.44246	0.0779995
$343$	1.00000	0.0539949
$344$	−26.5834	−1.43328
$345$	0	0
$346$	16.2870	0.875594
$347$	−3.88544	−0.208581	−0.104291	−	0.994547i	$-0.533257\pi$
$347$	−3.88544	−0.208581	−0.104291	+	0.994547i	$0.533257\pi$
$348$	0.284079	0.0152282
$349$	33.1267	1.77323	0.886616	−	0.462507i	$-0.153050\pi$
$349$	33.1267	1.77323	0.886616	+	0.462507i	$0.153050\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−16.2477	−0.866007
$353$	6.16721	0.328248	0.164124	−	0.986440i	$-0.447520\pi$
$353$	6.16721	0.328248	0.164124	+	0.986440i	$0.447520\pi$
$354$	−14.4691	−0.769025
$355$	0	0
$356$	9.45193	0.500951
$357$	3.24669	0.171833
$358$	−13.2017	−0.697731
$359$	−4.35926	−0.230073	−0.115036	−	0.993361i	$-0.536698\pi$
$359$	−4.35926	−0.230073	−0.115036	+	0.993361i	$0.536698\pi$
$360$	0	0
$361$	−18.2052	−0.958171
$362$	33.2520	1.74769
$363$	12.0806	0.634066
$364$	0.618034	0.0323938
$365$	0	0
$366$	16.6880	0.872296
$367$	−35.6921	−1.86311	−0.931555	−	0.363600i	$-0.881547\pi$
$367$	−35.6921	−1.86311	−0.931555	+	0.363600i	$0.881547\pi$
$368$	−6.85224	−0.357198
$369$	−3.07768	−0.160218
$370$	0	0
$371$	−0.405074	−0.0210304
$372$	0.210453	0.0109115
$373$	11.6759	0.604556	0.302278	−	0.953220i	$-0.402253\pi$
$373$	11.6759	0.604556	0.302278	+	0.953220i	$0.402253\pi$
$374$	25.2378	1.30502
$375$	0	0
$376$	−10.0836	−0.520022
$377$	0.459650	0.0236732
$378$	−1.61803	−0.0832227
$379$	20.0941	1.03217	0.516083	−	0.856539i	$-0.327390\pi$
$379$	20.0941	1.03217	0.516083	+	0.856539i	$0.327390\pi$
$380$	0	0
$381$	−13.3974	−0.686367
$382$	−23.9971	−1.22780
$383$	−0.874948	−0.0447078	−0.0223539	−	0.999750i	$-0.507116\pi$
$383$	−0.874948	−0.0447078	−0.0223539	+	0.999750i	$0.507116\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	7.09423	0.361087
$387$	−11.8885	−0.604325
$388$	−6.48276	−0.329112
$389$	32.7010	1.65801	0.829004	−	0.559243i	$-0.188908\pi$
$389$	32.7010	1.65801	0.829004	+	0.559243i	$0.188908\pi$
$390$	0	0
$391$	4.58315	0.231780
$392$	2.23607	0.112938
$393$	−20.5593	−1.03708
$394$	8.74258	0.440445
$395$	0	0
$396$	−2.96917	−0.149207
$397$	−37.1937	−1.86670	−0.933349	−	0.358971i	$-0.883128\pi$
$397$	−37.1937	−1.86670	−0.933349	+	0.358971i	$0.883128\pi$
$398$	18.7194	0.938316
$399$	−0.891491	−0.0446304
$400$	0	0
$401$	−3.69636	−0.184587	−0.0922937	−	0.995732i	$-0.529420\pi$
$401$	−3.69636	−0.184587	−0.0922937	+	0.995732i	$0.529420\pi$
$402$	−13.3385	−0.665266
$403$	0.340519	0.0169625
$404$	1.52130	0.0756875
$405$	0	0
$406$	−0.743729	−0.0369106
$407$	8.24367	0.408624
$408$	7.25982	0.359415
$409$	−5.50837	−0.272372	−0.136186	−	0.990683i	$-0.543484\pi$
$409$	−5.50837	−0.272372	−0.136186	+	0.990683i	$0.543484\pi$
$410$	0	0
$411$	−4.74258	−0.233934
$412$	−2.88564	−0.142165
$413$	8.94241	0.440027
$414$	−2.28408	−0.112256
$415$	0	0
$416$	3.38197	0.165815
$417$	14.9069	0.729993
$418$	−6.92992	−0.338953
$419$	−24.5229	−1.19802	−0.599012	−	0.800740i	$-0.704440\pi$
$419$	−24.5229	−1.19802	−0.599012	+	0.800740i	$0.704440\pi$
$420$	0	0
$421$	−6.91531	−0.337032	−0.168516	−	0.985699i	$-0.553897\pi$
$421$	−6.91531	−0.337032	−0.168516	+	0.985699i	$0.553897\pi$
$422$	17.4186	0.847924
$423$	−4.50953	−0.219261
$424$	−0.905773	−0.0439882
$425$	0	0
$426$	−12.2609	−0.594041
$427$	−10.3138	−0.499117
$428$	−3.88712	−0.187891
$429$	−4.80423	−0.231950
$430$	0	0
$431$	6.75428	0.325342	0.162671	−	0.986680i	$-0.447989\pi$
$431$	6.75428	0.325342	0.162671	+	0.986680i	$0.447989\pi$
$432$	−4.85410	−0.233543
$433$	26.2842	1.26314	0.631568	−	0.775320i	$-0.282412\pi$
$433$	26.2842	1.26314	0.631568	+	0.775320i	$0.282412\pi$
$434$	−0.550972	−0.0264475
$435$	0	0
$436$	−2.42297	−0.116039
$437$	−1.25846	−0.0602005
$438$	−4.01719	−0.191949
$439$	−21.6867	−1.03505	−0.517526	−	0.855668i	$-0.673147\pi$
$439$	−21.6867	−1.03505	−0.517526	+	0.855668i	$0.673147\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−5.25325	−0.249872
$443$	−30.0870	−1.42948	−0.714739	−	0.699391i	$-0.753455\pi$
$443$	−30.0870	−1.42948	−0.714739	+	0.699391i	$0.753455\pi$
$444$	−1.06050	−0.0503290
$445$	0	0
$446$	15.6294	0.740072
$447$	−9.09372	−0.430118
$448$	4.23607	0.200135
$449$	−37.4141	−1.76568	−0.882840	−	0.469673i	$-0.844372\pi$
$449$	−37.4141	−1.76568	−0.882840	+	0.469673i	$0.844372\pi$
$450$	0	0
$451$	14.7859	0.696240
$452$	−1.50702	−0.0708842
$453$	−12.9864	−0.610152
$454$	−45.0218	−2.11298
$455$	0	0
$456$	−1.99344	−0.0933512
$457$	−8.26452	−0.386598	−0.193299	−	0.981140i	$-0.561919\pi$
$457$	−8.26452	−0.386598	−0.193299	+	0.981140i	$0.561919\pi$
$458$	−13.8022	−0.644937
$459$	3.24669	0.151542
$460$	0	0
$461$	0.874815	0.0407442	0.0203721	−	0.999792i	$-0.493515\pi$
$461$	0.874815	0.0407442	0.0203721	+	0.999792i	$0.493515\pi$
$462$	7.77340	0.361651
$463$	−20.4850	−0.952020	−0.476010	−	0.879440i	$-0.657917\pi$
$463$	−20.4850	−0.952020	−0.476010	+	0.879440i	$0.657917\pi$
$464$	−2.23119	−0.103580
$465$	0	0
$466$	−34.2168	−1.58506
$467$	−12.7298	−0.589063	−0.294532	−	0.955642i	$-0.595164\pi$
$467$	−12.7298	−0.589063	−0.294532	+	0.955642i	$0.595164\pi$
$468$	0.618034	0.0285686
$469$	8.24367	0.380657
$470$	0	0
$471$	2.55703	0.117822
$472$	19.9958	0.920383
$473$	57.1149	2.62615
$474$	14.6774	0.674154
$475$	0	0
$476$	2.00656	0.0919707
$477$	−0.405074	−0.0185471
$478$	−29.4223	−1.34575
$479$	24.9962	1.14210	0.571052	−	0.820914i	$-0.306536\pi$
$479$	24.9962	1.14210	0.571052	+	0.820914i	$0.306536\pi$
$480$	0	0
$481$	−1.71592	−0.0782393
$482$	18.9345	0.862443
$483$	1.41164	0.0642318
$484$	7.46621	0.339373
$485$	0	0
$486$	−1.61803	−0.0733955
$487$	27.6813	1.25436	0.627180	−	0.778874i	$-0.284209\pi$
$487$	27.6813	1.25436	0.627180	+	0.778874i	$0.284209\pi$
$488$	−23.0622	−1.04398
$489$	25.2136	1.14020
$490$	0	0
$491$	5.75969	0.259931	0.129966	−	0.991518i	$-0.458513\pi$
$491$	5.75969	0.259931	0.129966	+	0.991518i	$0.458513\pi$
$492$	−1.90211	−0.0857539
$493$	1.49234	0.0672116
$494$	1.44246	0.0648995
$495$	0	0
$496$	−1.65292	−0.0742181
$497$	7.57763	0.339903
$498$	6.19764	0.277723
$499$	−35.2238	−1.57683	−0.788416	−	0.615143i	$-0.789098\pi$
$499$	−35.2238	−1.57683	−0.788416	+	0.615143i	$0.789098\pi$
$500$	0	0
$501$	−14.4068	−0.643646
$502$	−10.7467	−0.479651
$503$	−23.1576	−1.03254	−0.516272	−	0.856425i	$-0.672681\pi$
$503$	−23.1576	−1.03254	−0.516272	+	0.856425i	$0.672681\pi$
$504$	2.23607	0.0996024
$505$	0	0
$506$	10.9732	0.487820
$507$	1.00000	0.0444116
$508$	−8.28002	−0.367367
$509$	7.23722	0.320784	0.160392	−	0.987053i	$-0.448724\pi$
$509$	7.23722	0.320784	0.160392	+	0.987053i	$0.448724\pi$
$510$	0	0
$511$	2.48276	0.109831
$512$	5.29180	0.233867
$513$	−0.891491	−0.0393603
$514$	26.5937	1.17300
$515$	0	0
$516$	−7.34748	−0.323455
$517$	21.6648	0.952816
$518$	2.77642	0.121989
$519$	−10.0659	−0.441844
$520$	0	0
$521$	−16.2953	−0.713910	−0.356955	−	0.934122i	$-0.616185\pi$
$521$	−16.2953	−0.713910	−0.356955	+	0.934122i	$0.616185\pi$
$522$	−0.743729	−0.0325521
$523$	−22.1147	−0.967010	−0.483505	−	0.875342i	$-0.660637\pi$
$523$	−22.1147	−0.967010	−0.483505	+	0.875342i	$0.660637\pi$
$524$	−12.7063	−0.555079
$525$	0	0
$526$	45.1523	1.96873
$527$	1.10556	0.0481590
$528$	23.3202	1.01488
$529$	−21.0073	−0.913360
$530$	0	0
$531$	8.94241	0.388067
$532$	−0.550972	−0.0238877
$533$	−3.07768	−0.133309
$534$	−24.7455	−1.07084
$535$	0	0
$536$	18.4334	0.796202
$537$	8.15910	0.352091
$538$	−44.2999	−1.90990
$539$	−4.80423	−0.206933
$540$	0	0
$541$	−13.0517	−0.561138	−0.280569	−	0.959834i	$-0.590523\pi$
$541$	−13.0517	−0.561138	−0.280569	+	0.959834i	$0.590523\pi$
$542$	5.02562	0.215869
$543$	−20.5509	−0.881922
$544$	10.9802	0.470772
$545$	0	0
$546$	−1.61803	−0.0692455
$547$	−21.3283	−0.911933	−0.455966	−	0.889997i	$-0.650706\pi$
$547$	−21.3283	−0.911933	−0.455966	+	0.889997i	$0.650706\pi$
$548$	−2.93107	−0.125209
$549$	−10.3138	−0.440180
$550$	0	0
$551$	−0.409774	−0.0174569
$552$	3.15652	0.134350
$553$	−9.07112	−0.385743
$554$	−48.8882	−2.07706
$555$	0	0
$556$	9.21296	0.390717
$557$	18.4676	0.782496	0.391248	−	0.920285i	$-0.372043\pi$
$557$	18.4676	0.782496	0.391248	+	0.920285i	$0.372043\pi$
$558$	−0.550972	−0.0233245
$559$	−11.8885	−0.502829
$560$	0	0
$561$	−15.5978	−0.658541
$562$	1.49338	0.0629944
$563$	3.69475	0.155715	0.0778575	−	0.996964i	$-0.475192\pi$
$563$	3.69475	0.155715	0.0778575	+	0.996964i	$0.475192\pi$
$564$	−2.78704	−0.117356
$565$	0	0
$566$	27.1753	1.14226
$567$	1.00000	0.0419961
$568$	16.9441	0.710958
$569$	−31.4370	−1.31791	−0.658954	−	0.752183i	$-0.729001\pi$
$569$	−31.4370	−1.31791	−0.658954	+	0.752183i	$0.729001\pi$
$570$	0	0
$571$	32.3772	1.35494	0.677471	−	0.735549i	$-0.263076\pi$
$571$	32.3772	1.35494	0.677471	+	0.735549i	$0.263076\pi$
$572$	−2.96917	−0.124147
$573$	14.8310	0.619574
$574$	4.97980	0.207853
$575$	0	0
$576$	4.23607	0.176503
$577$	−26.6939	−1.11128	−0.555642	−	0.831422i	$-0.687527\pi$
$577$	−26.6939	−1.11128	−0.555642	+	0.831422i	$0.687527\pi$
$578$	10.4509	0.434700
$579$	−4.38447	−0.182212
$580$	0	0
$581$	−3.83035	−0.158910
$582$	16.9721	0.703515
$583$	1.94607	0.0805979
$584$	5.55161	0.229727
$585$	0	0
$586$	−8.94879	−0.369671
$587$	−25.0174	−1.03258	−0.516289	−	0.856414i	$-0.672687\pi$
$587$	−25.0174	−1.03258	−0.516289	+	0.856414i	$0.672687\pi$
$588$	0.618034	0.0254873
$589$	−0.303570	−0.0125084
$590$	0	0
$591$	−5.40321	−0.222258
$592$	8.32926	0.342330
$593$	−30.4293	−1.24958	−0.624791	−	0.780792i	$-0.714816\pi$
$593$	−30.4293	−1.24958	−0.624791	+	0.780792i	$0.714816\pi$
$594$	7.77340	0.318947
$595$	0	0
$596$	−5.62023	−0.230213
$597$	−11.5692	−0.473496
$598$	−2.28408	−0.0934029
$599$	−45.6159	−1.86382	−0.931908	−	0.362694i	$-0.881857\pi$
$599$	−45.6159	−1.86382	−0.931908	+	0.362694i	$0.881857\pi$
$600$	0	0
$601$	30.8212	1.25722	0.628611	−	0.777720i	$-0.283624\pi$
$601$	30.8212	1.25722	0.628611	+	0.777720i	$0.283624\pi$
$602$	19.2360	0.783999
$603$	8.24367	0.335708
$604$	−8.02601	−0.326574
$605$	0	0
$606$	−3.98281	−0.161791
$607$	−45.7895	−1.85854	−0.929269	−	0.369403i	$-0.879562\pi$
$607$	−45.7895	−1.85854	−0.929269	+	0.369403i	$0.879562\pi$
$608$	−3.01499	−0.122274
$609$	0.459650	0.0186259
$610$	0	0
$611$	−4.50953	−0.182436
$612$	2.00656	0.0811106
$613$	42.3482	1.71043	0.855214	−	0.518276i	$-0.173426\pi$
$613$	42.3482	1.71043	0.855214	+	0.518276i	$0.173426\pi$
$614$	50.7993	2.05009
$615$	0	0
$616$	−10.7426	−0.432831
$617$	20.2302	0.814438	0.407219	−	0.913331i	$-0.366499\pi$
$617$	20.2302	0.814438	0.407219	+	0.913331i	$0.366499\pi$
$618$	7.55470	0.303895
$619$	−42.9807	−1.72754	−0.863771	−	0.503885i	$-0.831903\pi$
$619$	−42.9807	−1.72754	−0.863771	+	0.503885i	$0.831903\pi$
$620$	0	0
$621$	1.41164	0.0566471
$622$	−4.42309	−0.177350
$623$	15.2935	0.612723
$624$	−4.85410	−0.194320
$625$	0	0
$626$	4.73258	0.189152
$627$	4.28293	0.171044
$628$	1.58033	0.0630620
$629$	−5.57106	−0.222133
$630$	0	0
$631$	−33.9944	−1.35329	−0.676647	−	0.736307i	$-0.736568\pi$
$631$	−33.9944	−1.35329	−0.676647	+	0.736307i	$0.736568\pi$
$632$	−20.2836	−0.806840
$633$	−10.7653	−0.427882
$634$	−1.31085	−0.0520604
$635$	0	0
$636$	−0.250349	−0.00992700
$637$	1.00000	0.0396214
$638$	3.57304	0.141458
$639$	7.57763	0.299766
$640$	0	0
$641$	−4.61668	−0.182348	−0.0911739	−	0.995835i	$-0.529062\pi$
$641$	−4.61668	−0.182348	−0.0911739	+	0.995835i	$0.529062\pi$
$642$	10.1766	0.401639
$643$	−15.0975	−0.595386	−0.297693	−	0.954662i	$-0.596217\pi$
$643$	−15.0975	−0.595386	−0.297693	+	0.954662i	$0.596217\pi$
$644$	0.872441	0.0343790
$645$	0	0
$646$	4.68323	0.184259
$647$	24.3079	0.955642	0.477821	−	0.878457i	$-0.341427\pi$
$647$	24.3079	0.955642	0.477821	+	0.878457i	$0.341427\pi$
$648$	2.23607	0.0878410
$649$	−42.9613	−1.68638
$650$	0	0
$651$	0.340519	0.0133460
$652$	15.5828	0.610271
$653$	36.8251	1.44108	0.720539	−	0.693415i	$-0.243895\pi$
$653$	36.8251	1.44108	0.720539	+	0.693415i	$0.243895\pi$
$654$	6.34342	0.248048
$655$	0	0
$656$	14.9394	0.583285
$657$	2.48276	0.0968616
$658$	7.29657	0.284450
$659$	26.7208	1.04090	0.520448	−	0.853893i	$-0.325765\pi$
$659$	26.7208	1.04090	0.520448	+	0.853893i	$0.325765\pi$
$660$	0	0
$661$	27.8739	1.08417	0.542084	−	0.840324i	$-0.317636\pi$
$661$	27.8739	1.08417	0.542084	+	0.840324i	$0.317636\pi$
$662$	8.12703	0.315866
$663$	3.24669	0.126091
$664$	−8.56493	−0.332384
$665$	0	0
$666$	2.77642	0.107584
$667$	0.648859	0.0251239
$668$	−8.90387	−0.344501
$669$	−9.65948	−0.373457
$670$	0	0
$671$	49.5496	1.91284
$672$	3.38197	0.130462
$673$	−2.24554	−0.0865591	−0.0432795	−	0.999063i	$-0.513781\pi$
$673$	−2.24554	−0.0865591	−0.0432795	+	0.999063i	$0.513781\pi$
$674$	24.2513	0.934124
$675$	0	0
$676$	0.618034	0.0237705
$677$	−3.64602	−0.140128	−0.0700640	−	0.997542i	$-0.522320\pi$
$677$	−3.64602	−0.140128	−0.0700640	+	0.997542i	$0.522320\pi$
$678$	3.94542	0.151523
$679$	−10.4893	−0.402543
$680$	0	0
$681$	27.8250	1.06626
$682$	2.64699	0.101359
$683$	−16.3180	−0.624391	−0.312196	−	0.950018i	$-0.601064\pi$
$683$	−16.3180	−0.624391	−0.312196	+	0.950018i	$0.601064\pi$
$684$	−0.550972	−0.0210669
$685$	0	0
$686$	−1.61803	−0.0617768
$687$	8.53026	0.325450
$688$	57.7079	2.20009
$689$	−0.405074	−0.0154321
$690$	0	0
$691$	25.9174	0.985946	0.492973	−	0.870045i	$-0.335910\pi$
$691$	25.9174	0.985946	0.492973	+	0.870045i	$0.335910\pi$
$692$	−6.22107	−0.236490
$693$	−4.80423	−0.182497
$694$	6.28677	0.238642
$695$	0	0
$696$	1.02781	0.0389589
$697$	−9.99228	−0.378485
$698$	−53.6001	−2.02879
$699$	21.1471	0.799858
$700$	0	0
$701$	3.27417	0.123664	0.0618318	−	0.998087i	$-0.480306\pi$
$701$	3.27417	0.123664	0.0618318	+	0.998087i	$0.480306\pi$
$702$	−1.61803	−0.0610688
$703$	1.52973	0.0576948
$704$	−20.3510	−0.767008
$705$	0	0
$706$	−9.97876	−0.375555
$707$	2.46151	0.0925748
$708$	5.52671	0.207707
$709$	−25.4583	−0.956105	−0.478052	−	0.878331i	$-0.658657\pi$
$709$	−25.4583	−0.956105	−0.478052	+	0.878331i	$0.658657\pi$
$710$	0	0
$711$	−9.07112	−0.340194
$712$	34.1974	1.28160
$713$	0.480690	0.0180020
$714$	−5.25325	−0.196598
$715$	0	0
$716$	5.04260	0.188451
$717$	18.1840	0.679094
$718$	7.05342	0.263231
$719$	−31.3560	−1.16938	−0.584690	−	0.811257i	$-0.698784\pi$
$719$	−31.3560	−1.16938	−0.584690	+	0.811257i	$0.698784\pi$
$720$	0	0
$721$	−4.66906	−0.173885
$722$	29.4567	1.09626
$723$	−11.7022	−0.435208
$724$	−12.7011	−0.472034
$725$	0	0
$726$	−19.5468	−0.725450
$727$	31.0101	1.15010	0.575050	−	0.818118i	$-0.304983\pi$
$727$	31.0101	1.15010	0.575050	+	0.818118i	$0.304983\pi$
$728$	2.23607	0.0828742
$729$	1.00000	0.0370370
$730$	0	0
$731$	−38.5982	−1.42761
$732$	−6.37425	−0.235599
$733$	−2.45901	−0.0908255	−0.0454127	−	0.998968i	$-0.514460\pi$
$733$	−2.45901	−0.0908255	−0.0454127	+	0.998968i	$0.514460\pi$
$734$	57.7510	2.13163
$735$	0	0
$736$	4.77411	0.175976
$737$	−39.6045	−1.45885
$738$	4.97980	0.183309
$739$	−8.78416	−0.323130	−0.161565	−	0.986862i	$-0.551654\pi$
$739$	−8.78416	−0.323130	−0.161565	+	0.986862i	$0.551654\pi$
$740$	0	0
$741$	−0.891491	−0.0327498
$742$	0.655423	0.0240614
$743$	−8.51627	−0.312432	−0.156216	−	0.987723i	$-0.549930\pi$
$743$	−8.51627	−0.312432	−0.156216	+	0.987723i	$0.549930\pi$
$744$	0.761425	0.0279152
$745$	0	0
$746$	−18.8920	−0.691686
$747$	−3.83035	−0.140145
$748$	−9.63999	−0.352473
$749$	−6.28949	−0.229813
$750$	0	0
$751$	33.0460	1.20587	0.602933	−	0.797791i	$-0.293998\pi$
$751$	33.0460	1.20587	0.602933	+	0.797791i	$0.293998\pi$
$752$	21.8897	0.798235
$753$	6.64185	0.242043
$754$	−0.743729	−0.0270850
$755$	0	0
$756$	0.618034	0.0224777
$757$	−29.7676	−1.08192	−0.540961	−	0.841048i	$-0.681939\pi$
$757$	−29.7676	−1.08192	−0.540961	+	0.841048i	$0.681939\pi$
$758$	−32.5130	−1.18092
$759$	−6.78183	−0.246165
$760$	0	0
$761$	14.2403	0.516209	0.258104	−	0.966117i	$-0.416902\pi$
$761$	14.2403	0.516209	0.258104	+	0.966117i	$0.416902\pi$
$762$	21.6774	0.785289
$763$	−3.92045	−0.141930
$764$	9.16606	0.331616
$765$	0	0
$766$	1.41570	0.0511512
$767$	8.94241	0.322892
$768$	13.5623	0.489388
$769$	−38.0129	−1.37078	−0.685390	−	0.728176i	$-0.740368\pi$
$769$	−38.0129	−1.37078	−0.685390	+	0.728176i	$0.740368\pi$
$770$	0	0
$771$	−16.4358	−0.591920
$772$	−2.70975	−0.0975262
$773$	39.9492	1.43687	0.718436	−	0.695593i	$-0.244858\pi$
$773$	39.9492	1.43687	0.718436	+	0.695593i	$0.244858\pi$
$774$	19.2360	0.691422
$775$	0	0
$776$	−23.4548	−0.841980
$777$	−1.71592	−0.0615583
$778$	−52.9114	−1.89696
$779$	2.74373	0.0983043
$780$	0	0
$781$	−36.4046	−1.30266
$782$	−7.41570	−0.265185
$783$	0.459650	0.0164265
$784$	−4.85410	−0.173361
$785$	0	0
$786$	33.2656	1.18655
$787$	−40.2577	−1.43503	−0.717517	−	0.696541i	$-0.754721\pi$
$787$	−40.2577	−1.43503	−0.717517	+	0.696541i	$0.754721\pi$
$788$	−3.33937	−0.118960
$789$	−27.9057	−0.993468
$790$	0	0
$791$	−2.43841	−0.0866998
$792$	−10.7426	−0.381721
$793$	−10.3138	−0.366252
$794$	60.1806	2.13573
$795$	0	0
$796$	−7.15016	−0.253431
$797$	15.9645	0.565493	0.282747	−	0.959195i	$-0.408754\pi$
$797$	15.9645	0.565493	0.282747	+	0.959195i	$0.408754\pi$
$798$	1.44246	0.0510627
$799$	−14.6410	−0.517962
$800$	0	0
$801$	15.2935	0.540371
$802$	5.98084	0.211191
$803$	−11.9277	−0.420920
$804$	5.09487	0.179682
$805$	0	0
$806$	−0.550972	−0.0194072
$807$	27.3788	0.963781
$808$	5.50411	0.193634
$809$	−31.1877	−1.09650	−0.548251	−	0.836314i	$-0.684706\pi$
$809$	−31.1877	−1.09650	−0.548251	+	0.836314i	$0.684706\pi$
$810$	0	0
$811$	43.7661	1.53684	0.768418	−	0.639948i	$-0.221044\pi$
$811$	43.7661	1.53684	0.768418	+	0.639948i	$0.221044\pi$
$812$	0.284079	0.00996922
$813$	−3.10600	−0.108932
$814$	−13.3385	−0.467516
$815$	0	0
$816$	−15.7598	−0.551702
$817$	10.5985	0.370794
$818$	8.91273	0.311626
$819$	1.00000	0.0349428
$820$	0	0
$821$	−29.9442	−1.04506	−0.522529	−	0.852621i	$-0.675011\pi$
$821$	−29.9442	−1.04506	−0.522529	+	0.852621i	$0.675011\pi$
$822$	7.67365	0.267649
$823$	−10.1764	−0.354728	−0.177364	−	0.984145i	$-0.556757\pi$
$823$	−10.1764	−0.354728	−0.177364	+	0.984145i	$0.556757\pi$
$824$	−10.4403	−0.363707
$825$	0	0
$826$	−14.4691	−0.503445
$827$	−36.8304	−1.28072	−0.640360	−	0.768075i	$-0.721215\pi$
$827$	−36.8304	−1.28072	−0.640360	+	0.768075i	$0.721215\pi$
$828$	0.872441	0.0303194
$829$	18.6901	0.649133	0.324567	−	0.945863i	$-0.394782\pi$
$829$	18.6901	0.649133	0.324567	+	0.945863i	$0.394782\pi$
$830$	0	0
$831$	30.2146	1.04813
$832$	4.23607	0.146859
$833$	3.24669	0.112491
$834$	−24.1198	−0.835202
$835$	0	0
$836$	2.64699	0.0915482
$837$	0.340519	0.0117701
$838$	39.6789	1.37069
$839$	42.0400	1.45138	0.725692	−	0.688020i	$-0.241520\pi$
$839$	42.0400	1.45138	0.725692	+	0.688020i	$0.241520\pi$
$840$	0	0
$841$	−28.7887	−0.992715
$842$	11.1892	0.385606
$843$	−0.922959	−0.0317884
$844$	−6.65331	−0.229016
$845$	0	0
$846$	7.29657	0.250861
$847$	12.0806	0.415094
$848$	1.96627	0.0675220
$849$	−16.7953	−0.576413
$850$	0	0
$851$	−2.42226	−0.0830340
$852$	4.68323	0.160445
$853$	−33.0029	−1.13000	−0.564998	−	0.825092i	$-0.691123\pi$
$853$	−33.0029	−1.13000	−0.564998	+	0.825092i	$0.691123\pi$
$854$	16.6880	0.571052
$855$	0	0
$856$	−14.0637	−0.480688
$857$	−6.77951	−0.231583	−0.115792	−	0.993274i	$-0.536941\pi$
$857$	−6.77951	−0.231583	−0.115792	+	0.993274i	$0.536941\pi$
$858$	7.77340	0.265380
$859$	−47.5032	−1.62079	−0.810395	−	0.585884i	$-0.800747\pi$
$859$	−47.5032	−1.62079	−0.810395	+	0.585884i	$0.800747\pi$
$860$	0	0
$861$	−3.07768	−0.104887
$862$	−10.9287	−0.372231
$863$	−35.7993	−1.21862	−0.609311	−	0.792931i	$-0.708554\pi$
$863$	−35.7993	−1.21862	−0.609311	+	0.792931i	$0.708554\pi$
$864$	3.38197	0.115057
$865$	0	0
$866$	−42.5287	−1.44518
$867$	−6.45901	−0.219359
$868$	0.210453	0.00714323
$869$	43.5797	1.47834
$870$	0	0
$871$	8.24367	0.279326
$872$	−8.76640	−0.296868
$873$	−10.4893	−0.355010
$874$	2.03624	0.0688767
$875$	0	0
$876$	1.53443	0.0518435
$877$	0.361560	0.0122090	0.00610451	−	0.999981i	$-0.498057\pi$
$877$	0.361560	0.0122090	0.00610451	+	0.999981i	$0.498057\pi$
$878$	35.0899	1.18423
$879$	5.53066	0.186544
$880$	0	0
$881$	−15.8057	−0.532506	−0.266253	−	0.963903i	$-0.585786\pi$
$881$	−15.8057	−0.532506	−0.266253	+	0.963903i	$0.585786\pi$
$882$	−1.61803	−0.0544820
$883$	16.0987	0.541765	0.270883	−	0.962612i	$-0.412684\pi$
$883$	16.0987	0.541765	0.270883	+	0.962612i	$0.412684\pi$
$884$	2.00656	0.0674881
$885$	0	0
$886$	48.6819	1.63550
$887$	23.3425	0.783764	0.391882	−	0.920015i	$-0.371824\pi$
$887$	23.3425	0.783764	0.391882	+	0.920015i	$0.371824\pi$
$888$	−3.83692	−0.128758
$889$	−13.3974	−0.449333
$890$	0	0
$891$	−4.80423	−0.160948
$892$	−5.96989	−0.199887
$893$	4.02020	0.134531
$894$	14.7139	0.492108
$895$	0	0
$896$	−13.6180	−0.454947
$897$	1.41164	0.0471332
$898$	60.5373	2.02016
$899$	0.156520	0.00522022
$900$	0	0
$901$	−1.31515	−0.0438140
$902$	−23.9241	−0.796584
$903$	−11.8885	−0.395624
$904$	−5.45244	−0.181346
$905$	0	0
$906$	21.0124	0.698089
$907$	43.6032	1.44782	0.723910	−	0.689894i	$-0.242343\pi$
$907$	43.6032	1.44782	0.723910	+	0.689894i	$0.242343\pi$
$908$	17.1968	0.570696
$909$	2.46151	0.0816433
$910$	0	0
$911$	−42.3998	−1.40477	−0.702385	−	0.711798i	$-0.747881\pi$
$911$	−42.3998	−1.40477	−0.702385	+	0.711798i	$0.747881\pi$
$912$	4.32739	0.143294
$913$	18.4019	0.609013
$914$	13.3723	0.442315
$915$	0	0
$916$	5.27199	0.174191
$917$	−20.5593	−0.678928
$918$	−5.25325	−0.173383
$919$	−6.50432	−0.214558	−0.107279	−	0.994229i	$-0.534214\pi$
$919$	−6.50432	−0.214558	−0.107279	+	0.994229i	$0.534214\pi$
$920$	0	0
$921$	−31.3957	−1.03452
$922$	−1.41548	−0.0466164
$923$	7.57763	0.249421
$924$	−2.96917	−0.0976787
$925$	0	0
$926$	33.1455	1.08923
$927$	−4.66906	−0.153352
$928$	1.55452	0.0510296
$929$	44.7854	1.46936	0.734681	−	0.678412i	$-0.237332\pi$
$929$	44.7854	1.46936	0.734681	+	0.678412i	$0.237332\pi$
$930$	0	0
$931$	−0.891491	−0.0292175
$932$	13.0697	0.428111
$933$	2.73362	0.0894946
$934$	20.5972	0.673961
$935$	0	0
$936$	2.23607	0.0730882
$937$	9.80796	0.320412	0.160206	−	0.987084i	$-0.448784\pi$
$937$	9.80796	0.320412	0.160206	+	0.987084i	$0.448784\pi$
$938$	−13.3385	−0.435519
$939$	−2.92489	−0.0954503
$940$	0	0
$941$	6.56164	0.213903	0.106952	−	0.994264i	$-0.465891\pi$
$941$	6.56164	0.213903	0.106952	+	0.994264i	$0.465891\pi$
$942$	−4.13736	−0.134802
$943$	−4.34458	−0.141479
$944$	−43.4074	−1.41279
$945$	0	0
$946$	−92.4139	−3.00463
$947$	19.7936	0.643206	0.321603	−	0.946875i	$-0.395778\pi$
$947$	19.7936	0.643206	0.321603	+	0.946875i	$0.395778\pi$
$948$	−5.60626	−0.182083
$949$	2.48276	0.0805937
$950$	0	0
$951$	0.810148	0.0262708
$952$	7.25982	0.235292
$953$	15.0734	0.488275	0.244138	−	0.969741i	$-0.421495\pi$
$953$	15.0734	0.488275	0.244138	+	0.969741i	$0.421495\pi$
$954$	0.655423	0.0212201
$955$	0	0
$956$	11.2383	0.363474
$957$	−2.20826	−0.0713829
$958$	−40.4446	−1.30671
$959$	−4.74258	−0.153146
$960$	0	0
$961$	−30.8840	−0.996260
$962$	2.77642	0.0895153
$963$	−6.28949	−0.202676
$964$	−7.23234	−0.232938
$965$	0	0
$966$	−2.28408	−0.0734890
$967$	2.83139	0.0910514	0.0455257	−	0.998963i	$-0.485504\pi$
$967$	2.83139	0.0910514	0.0455257	+	0.998963i	$0.485504\pi$
$968$	27.0130	0.868231
$969$	−2.89440	−0.0929814
$970$	0	0
$971$	−2.43707	−0.0782093	−0.0391047	−	0.999235i	$-0.512451\pi$
$971$	−2.43707	−0.0782093	−0.0391047	+	0.999235i	$0.512451\pi$
$972$	0.618034	0.0198234
$973$	14.9069	0.477893
$974$	−44.7893	−1.43514
$975$	0	0
$976$	50.0640	1.60251
$977$	−51.3238	−1.64199	−0.820996	−	0.570934i	$-0.806581\pi$
$977$	−51.3238	−1.64199	−0.820996	+	0.570934i	$0.806581\pi$
$978$	−40.7964	−1.30452
$979$	−73.4737	−2.34823
$980$	0	0
$981$	−3.92045	−0.125170
$982$	−9.31938	−0.297393
$983$	10.8569	0.346283	0.173141	−	0.984897i	$-0.444608\pi$
$983$	10.8569	0.346283	0.173141	+	0.984897i	$0.444608\pi$
$984$	−6.88191	−0.219387
$985$	0	0
$986$	−2.41466	−0.0768983
$987$	−4.50953	−0.143540
$988$	−0.550972	−0.0175288
$989$	−16.7822	−0.533644
$990$	0	0
$991$	−5.95012	−0.189012	−0.0945060	−	0.995524i	$-0.530127\pi$
$991$	−5.95012	−0.189012	−0.0945060	+	0.995524i	$0.530127\pi$
$992$	1.15163	0.0365641
$993$	−5.02278	−0.159393
$994$	−12.2609	−0.388891
$995$	0	0
$996$	−2.36729	−0.0750104
$997$	3.66128	0.115954	0.0579769	−	0.998318i	$-0.481535\pi$
$997$	3.66128	0.115954	0.0579769	+	0.998318i	$0.481535\pi$
$998$	56.9932	1.80409
$999$	−1.71592	−0.0542893

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6825.2.a.be.1.1		4
5.2	odd	4		1365.2.f.c.274.1	✓	8
5.3	odd	4		1365.2.f.c.274.7	yes	8
5.4	even	2		6825.2.a.bm.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1365.2.f.c.274.1	✓	8	5.2	odd	4
1365.2.f.c.274.7	yes	8	5.3	odd	4
6825.2.a.be.1.1		4	1.1	even	1	trivial
6825.2.a.bm.1.3		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6825.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} +1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -4.80423 q^{11} +0.618034 q^{12} +1.00000 q^{13} -1.61803 q^{14} -4.85410 q^{16} +3.24669 q^{17} -1.61803 q^{18} -0.891491 q^{19} +1.00000 q^{21} +7.77340 q^{22} +1.41164 q^{23} +2.23607 q^{24} -1.61803 q^{26} +1.00000 q^{27} +0.618034 q^{28} +0.459650 q^{29} +0.340519 q^{31} +3.38197 q^{32} -4.80423 q^{33} -5.25325 q^{34} +0.618034 q^{36} -1.71592 q^{37} +1.44246 q^{38} +1.00000 q^{39} -3.07768 q^{41} -1.61803 q^{42} -11.8885 q^{43} -2.96917 q^{44} -2.28408 q^{46} -4.50953 q^{47} -4.85410 q^{48} +1.00000 q^{49} +3.24669 q^{51} +0.618034 q^{52} -0.405074 q^{53} -1.61803 q^{54} +2.23607 q^{56} -0.891491 q^{57} -0.743729 q^{58} +8.94241 q^{59} -10.3138 q^{61} -0.550972 q^{62} +1.00000 q^{63} +4.23607 q^{64} +7.77340 q^{66} +8.24367 q^{67} +2.00656 q^{68} +1.41164 q^{69} +7.57763 q^{71} +2.23607 q^{72} +2.48276 q^{73} +2.77642 q^{74} -0.550972 q^{76} -4.80423 q^{77} -1.61803 q^{78} -9.07112 q^{79} +1.00000 q^{81} +4.97980 q^{82} -3.83035 q^{83} +0.618034 q^{84} +19.2360 q^{86} +0.459650 q^{87} -10.7426 q^{88} +15.2935 q^{89} +1.00000 q^{91} +0.872441 q^{92} +0.340519 q^{93} +7.29657 q^{94} +3.38197 q^{96} -10.4893 q^{97} -1.61803 q^{98} -4.80423 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{3} - 2 q^{4} - 2 q^{6} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 6 q^{16} - 2 q^{17} - 2 q^{18} - 2 q^{19} + 4 q^{21} + 2 q^{22} - 8 q^{23} - 2 q^{26} + 4 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^3 - 2 * q^4 - 2 * q^6 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 6 * q^16 - 2 * q^17 - 2 * q^18 - 2 * q^19 + 4 * q^21 + 2 * q^22 - 8 * q^23 - 2 * q^26 + 4 * q^27 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 18 * q^32 - 4 * q^33 - 4 * q^34 - 2 * q^36 - 10 * q^37 + 6 * q^38 + 4 * q^39 - 2 * q^42 - 18 * q^43 + 2 * q^44 - 6 * q^46 - 12 * q^47 - 6 * q^48 + 4 * q^49 - 2 * q^51 - 2 * q^52 + 10 * q^53 - 2 * q^54 - 2 * q^57 + 8 * q^58 + 4 * q^59 - 20 * q^61 - 4 * q^62 + 4 * q^63 + 8 * q^64 + 2 * q^66 - 10 * q^67 + 6 * q^68 - 8 * q^69 - 14 * q^71 - 14 * q^73 + 10 * q^74 - 4 * q^76 - 4 * q^77 - 2 * q^78 - 26 * q^79 + 4 * q^81 - 12 * q^83 - 2 * q^84 + 4 * q^86 - 6 * q^87 + 20 * q^89 + 4 * q^91 + 14 * q^92 - 2 * q^93 + 16 * q^94 + 18 * q^96 - 16 * q^97 - 2 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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